Non-standard central limit conjecture for the edge-triangle model

Let $S_n$ be the number of non-zero elements of the random adjacency matrix under $\mathbb{P}_{n;\alpha,h}$, let $(\alpha_c,h_c)$ be the critical point, and let $u^*(\alpha_c,h_c)$ denote the corresponding limiting edge density. Let $Y$ be a random variable with Lebesgue density proportional to $e^{-\frac{81}{64}y^4}$. Non-standard central limit conjecture. At the critical point $(\alpha,h)=(\alpha_c,h_c)$, one has

$$2\,\frac{S_n-\frac{n^2}{2}u^*(\alpha_c,h_c)}{n^{3/2}}\xrightarrow{\;\mathrm{d}\;}Y$$

with respect to $\mathbb{P}_{n;\alpha_c,h_c}$ as $n\to+\infty$. The conjecture predicts critical fluctuations in the edge-triangle model matching those of its mean-field approximation; technical difficulties prevent the authors from proving this result.